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GRE Prep Club Team Member
Joined: 20 Feb 2017
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The shaded region in the figure above represents a rectangular frame
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20 Sep 2021, 04:53
20 Sep 2021, 04:53
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The shaded region in the figure above represents a rectangular frame with length 18 inches and width 15 inches. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the lenght and width of the frame, what is the length of the picture, in inches?
A. \(9\sqrt2\)
B. \(\frac {3}{2}\)
C. \(\frac {9}{\sqrt2}\)
D. \(15 ( 1 - \frac {1}{\sqrt2})\)
E. \(\frac {9}{2}\)
Re: The shaded region in the figure above represents a rectangular frame
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20 Sep 2021, 07:40
20 Sep 2021, 07:40
1
Let the internal rectangle be of Length L and Width(W).
So given Area of shaded is = area of internal rectangle , \(18 * 15 = 2 * L * W\).
Draw a diagonal and there will be 2 similar triangles, so \(\frac{18}{15} = \frac{L}{W}\).
Solve for L and it will be \(9\sqrt{2}\)
So given Area of shaded is = area of internal rectangle , \(18 * 15 = 2 * L * W\).
Draw a diagonal and there will be 2 similar triangles, so \(\frac{18}{15} = \frac{L}{W}\).
Solve for L and it will be \(9\sqrt{2}\)
Re: The shaded region in the figure above represents a rectangular frame
[#permalink]
22 Nov 2021, 02:49
22 Nov 2021, 02:49
1
Let \(l\) and \(w\) be the length and width of picture respectively that is being enclosed within the picture
Length and width of frame is given as \(18\) and \(15\) respectively.
length and width of the picture have the same ratio as the length and width of the frame, which means \(\frac{l}{w} = \frac{18}{15} = \frac{6}{5} \Rightarrow w = \frac{5}{6}l\)
Area of the frame itself = Total area of entire square - Area of the picture = \(18*15 - l*b\)
The frame encloses a rectangular picture that has the same area as the frame itself, meaning
\(18*15 - lb = lb\)
\(18*15 = 2lb = 2(l)(\frac{5}{6}l) = \frac{5}{3}l^2\)
\(l^2 = \frac{18*15*3}{5} = 18*3*3 = 9*2*9 = 9^2*2\)
\(l = \sqrt{9^2*2} = 9\sqrt{2}\)
Hence, Answer is A
Length and width of frame is given as \(18\) and \(15\) respectively.
length and width of the picture have the same ratio as the length and width of the frame, which means \(\frac{l}{w} = \frac{18}{15} = \frac{6}{5} \Rightarrow w = \frac{5}{6}l\)
Area of the frame itself = Total area of entire square - Area of the picture = \(18*15 - l*b\)
The frame encloses a rectangular picture that has the same area as the frame itself, meaning
\(18*15 - lb = lb\)
\(18*15 = 2lb = 2(l)(\frac{5}{6}l) = \frac{5}{3}l^2\)
\(l^2 = \frac{18*15*3}{5} = 18*3*3 = 9*2*9 = 9^2*2\)
\(l = \sqrt{9^2*2} = 9\sqrt{2}\)
Hence, Answer is A
